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From the granular Leidenfrost state to buoyancy-driven convection

Nicolas Rivas, Anthony R. Thornton, Stefan Luding, and Devaraj van der Meer
Phys. Rev. E 91, 042202 – Published 23 April 2015
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  • INTRODUCTION
  • SYSTEM AND SIMULATIONS
  • RESULTS
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
    • References

    Abstract

    Grains inside a vertically vibrated box undergo a transition from a density-inverted and horizontally homogeneous state, referred to as the granular Leidenfrost state, to a buoyancy-driven convective state. We perform a simulational study of the precursors of such a transition and quantify their dynamics as the bed of grains is progressively fluidized. The transition is preceded by transient convective states, which increase their correlation time as the transition point is approached. Increasingly correlated convective flows lead to density fluctuations, as quantified by the structure factor, that also shows critical behavior near the transition point. The amplitude of the modulations in the vertical velocity field are seen to be best described by a quintic supercritical amplitude equation with an additive noise term. The validity of such an amplitude equation, and previously observed collective semiperiodic oscillations of the bed of grains, suggests a new interpretation of the transition analogous to a coupled chain of vertically vibrated damped oscillators. Increasing the size of the container shows metastability of convective states, as well as an overall invariant critical behavior close to the transition.

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    • Received 6 January 2015

    DOI:https://doi.org/10.1103/PhysRevE.91.042202

    ©2015 American Physical Society

    Authors & Affiliations

    Nicolas Rivas1, Anthony R. Thornton1,2, Stefan Luding1, and Devaraj van der Meer3

    • 1Multi-Scale Mechanics (MSM), MESA +, CTW, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
    • 2Mathematics of Computational Science (MaCS), MESA +, CTW, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
    • 3Physics of Fluids, University of Twente, The Netherlands

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    Issue

    Vol. 91, Iss. 4 — April 2015

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    Images

    • Figure 1
      Figure 1

      Schematic representation (not to scale) of the setup. Two different geometries are considered: narrow (top) and wide (bottom). Lengths are given in units of particle diameters d.

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    • Figure 2
      Figure 2

      In the narrow system, time-averaged number density of particles 〈n(x,z)〉t (top), granular temperature 〈T(x,z)〉t (middle), and velocity field 〈v⃗(x,z)〉t (bottom) for systems in the granular Leidenfrost state (left) and in the buoyancy-driven convective state (right).

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    • Figure 3
      Figure 3

      Time-averaged convection intensity 〈C〉t, defined in the main text (1) as a function of the angular driving frequency ω, for the narrow (top) and wide (bottom) containers with the boundary conditions indicated in the labels. The vertical lines indicate the transition region for the corresponding BC, as specified in the main text. The thick solid line corresponds to Aω, the characteristic shaking velocity. The insets show the convection intensity normalized by the driving frequency, C*≡〈C〉t/Aω, as a function of the bifurcation parameter ɛ=(ω−ωc)/ωc.

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    • Figure 4
      Figure 4

      Transient velocity fields for ɛ=−0.15, each averaged over five oscillation periods, showing the emergence and decay of a fluctuating convective cell in a section of a wide container. From yellow (white) to purple (black), the color and size of the vectors corresponds to their norm.

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    • Figure 5
      Figure 5

      (a) Velocity correlation functions Fv for several ω and EBC in the narrow container. (b) Characteristic time-scale of fluctuating convection τv, corresponding to the exponent of the long term exponential decay of the self correlation function Fv, as a function of the bifurcation parameter ɛ. The dashed lines indicate best fits of the form indicated in the main text.

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    • Figure 6
      Figure 6

      Structure factor, S(k), for narrow (left) and wide (right) containers for the bifurcation parameters specified. Light colors (light gray) lines correspond to PBC, while dark colors (dark gray) lines have EBC. The vertical solid line indicates the 1/lx point.

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    • Figure 7
      Figure 7

      The most unstable mode kc, defined by the maximum of the structure factor max(S(k))≡S(kc), as a function of the bifurcation parameter for narrow (left) and wide (right) containers and the boundary conditions specified. The inset shows the convective length scale λc as a function of the shaking strength, as defined in the main text.

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    • Figure 8
      Figure 8

      Structure factor maximum, Sm, as a function of the bifurcation parameter ɛ for EBC (circles) and PBC (squares) in narrow (top) and wide (bottom) containers. As a reference, the best fit for the amplitude of the critical mode is included (dashed gray, see main text).

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    • Figure 9
      Figure 9

      Spatiotemporal contours plot of the number of particles field, n(x,t), for (a) lx=50, (b) lx=80, and (c) lx=400, with ω=35 (ɛ∼0.06) and EBC. These correspond to lx/λc*∼1,lx/λc*∼1.6, and lx/λc*∼8, respectively. High-density regions are shown in purple (black). Over the middle figure, the number of convection rolls is indicated for exemplary regions.

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    • Figure 10
      Figure 10

      Amplitude of the critical pattern of the vertical velocity field vz(x,t),Ac, as a function of the bifurcation parameter ɛ, for EBC (circles) and PBC (squares) in narrow (top) and wide (bottom) containers. The dashed lines correspond to fits given by the Swift-Hohenberg model with a stochastic term (see main text), with noise level η=0.0001. The solid lines correspond to fits based on a quintic supercritical bifurcation, for noise intensity σ=0.0008 in the small container systems and σ=0.001 for the wide cases.

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